2.48Definition (Field.)
A field is a triple where is a set, and and are
binary operations on (called addition and multiplication
respectively) satisfying the following nine conditions. (These conditions are called
the field axioms.)

(Associativity of addition.)Addition is an associative operation
on .

(Existence of additive identity.)There is an identity element for
addition.

We know that this identity is unique, and we will denote it by .

(Existence of additive inverses.)Every element of is invertible
for .

We know that the additive inverse for is unique, and we will denote it
by .

(Commutativity of multiplication.)Multiplication is a
commutative operation on .

(Associativity of multiplication.)Multiplication is an associative
operation on .

(Existence of multiplicative identity.)There is an identity element for
multiplication.

We know that this identity is unique, and we will denote it by .

(Existence of multiplicative inverses.)Every element of except possibly for is invertible for .

We know that the multiplicative
inverse for is unique, and we will denote it by .
We do not
assume is not invertible. We just do not assume that it is.

2.49Remark.
Most calculus books that begin with the axioms for a field (e.g.,
[47, p5], [1, p18], [13, p5], [12, p554])
add an additional axiom.

10.

(Commutativity of addition.)Addition is a commutative operation on
.

I have omitted this because, as Leonard Dickson
pointed out in 1905[18, p202], it
can be proved from the other axioms (see theorem 2.72 for a proof). I
agree with Aristotle that

It is manifest that it is far better to make the principles finite in
number. Nay, they should be the fewest possible provided they enable all the same
results to be proved. This is what mathematicians insist upon; for they take as
principles things finite either in kind or in number[26, p178].

2.50Remark (Parentheses.)The distributive law is usually written as

(2.51)

The right side of (2.51) is ambiguous. There are five sensible ways to
interpret it:

The conventions presently used for interpreting ambiguous statements such as
and involving operations are:

Multiplication and division have equal precedence.

Addition and subtraction have equal precedence.

Multiplication has higher precedence than addition.

This means that to interpret

(2.52)

you first read (2.52) from left to right and perform all the multipliations and
divisions as you come to them, getting

(2.53)

Then read (2.53) from left to right performing all additions and subtractions
as you come to them, getting

When I was in high school, multiplication had higher precedence than division, so

meant

whereas today it means

In 1713, addition often had higher precedence than multiplication.
Jacob Bernoulli
[8, p180] wrote expressions like

to mean

2.54Examples. with the usual operations of addition and multiplication is a field.

is a field. (See definition 2.42 for the
definitions.) We showed in section 2.2 that
satisfies all the field axioms except possibly the distributive law. In appendix
B, it is shown that the distributive property holds for
for all
, . (The proof assumes that the
distributive law holds in
.)

For a general
, , the only field axiom that can possibly fail to
hold in
is the existence of multiplicative inverses, so to
determine whether
is a field, it is just necessary to determine whether every
non-zero element in
is invertible for .

2.55Exercise.A
In each of the examples
below, determine which field axioms are
valid and which are not. Which examples are fields? In each case that an axiom
fails to hold, give an example to show why it fails to hold.

a)
where and are usual addition and
multiplication.

b) where
is the set of non-negative rational
numbers, and and are the usual addition and multiplication.

c) where is a set with just one element and both
and are the only binary operation on ; i.e.,

d)
where both and are the usual
operation of addition on
, e.g., and .

2.56Exercise.
Determine for which values of ,
is a
field. (You already know that produces a field.)

2.57Notation (The field
.)
Let
, be a number such that
is a field.
Then `` the field
" means the field
. I will
often denote the operations in
by and instead of and
.

2.58Entertainment.
Determine for which values of in
the system
is a field. If you do this you will probably conjecture
the exact (fairly simple) condition on that makes the system into a field.